104 lines
3.8 KiB
Markdown
104 lines
3.8 KiB
Markdown
+++
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title = "Signal to Noise Ratio"
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author = ["Thomas Dehaeze"]
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draft = false
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Tags
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: [Electronics]({{< relref "electronics" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
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## SNR to Noise PSD {#snr-to-noise-psd}
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From ([Jabben 2007](#orgf2f4e47)) (Section 3.3.2):
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> Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
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> In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
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> The noise performance of components like sensors, amplifiers, converters, etc., is often specified in terms of a **Signal to Noise Ratio** (SNR).
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> The SNR gives the ratio of the RMS value of a sine that covers the full range of the channel through which the signal is propagating over the RMS value of the electrical noise.
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>
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> Usually, the SNR is specified up to a certain cut-off frequency.
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> If no information on the colouring of the noise is available, then the corresponding **PSD can be assumed to be white up to the cut-off frequency** \\(f\_c\\):
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> \\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
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> with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.
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<div class="exampl">
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<div></div>
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Let's take an example.
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- \\(x\_{fr} = 170 V\\)
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- \\(C\_{snr} = 85 dB\\)
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- \\(f\_c = 200 Hz\\)
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The Power Spectral Density of the output voltage is:
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\\[ S\_{snr} = \frac{170^2}{8 \cdot 200 \cdot {10^{\frac{2 \cdot 85}{20}}}} = 5.7 \cdot 10^{-8}\ V^2/Hz \\]
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And the RMS of that noise up to \\(f\_c\\) is:
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\\[ S\_{rms} = \sqrt{S\_{snr} \cdot f\_c} \approx 3.4\ mV \\]
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</div>
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## Convert SNR to Noise RMS value {#convert-snr-to-noise-rms-value}
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The RMS value of the noise can be computed from:
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\\[ N\_\text{rms} = 10^{-\frac{S\_{snr}}{20}} S\_\text{rms} \\]
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where \\(S\_{snr}\\) is the SNR in dB and \\(S\_\text{rms}\\) is the RMS value of a sinus taking the full range.
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If the full range is \\(\Delta V\\), then:
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\\[ S\_\text{rms} = \frac{\Delta V/2}{\sqrt{2}} \\]
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<div class="exampl">
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<div></div>
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As an example, let's take a voltage amplifier with a full range of \\(\Delta V = 20V\\) and a SNR of 85dB.
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The RMS value of the noise is then:
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\\[ n\_\text{rms} = 10^{-\frac{S\_{nrs}}{20}} s\_\text{rms} \\]
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\\[ n\_\text{rms} = 10^{-\frac{85}{20}} \frac{10}{\sqrt{2}} \approx 0.4 mV\_\text{rms} \\]
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</div>
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## Convert wanted Noise RMS value to required SNR {#convert-wanted-noise-rms-value-to-required-snr}
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If the wanted full range and RMS value of the noise are defined, the required SNR can be computed from:
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\\[ S\_{snr} = 20 \log \frac{\text{Signal, rms}}{\text{Noise, rms}} \\]
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<div class="exampl">
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<div></div>
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Let's say the wanted noise is \\(1 mV, \text{rms}\\) for a full range of \\(20 V\\), the corresponding SNR is:
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\\[ S\_{snr} = 20 \log \frac{\frac{20/2}{\sqrt{2}}}{10^{-3}} \approx 77dB \\]
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</div>
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## Noise Density to RMS noise {#noise-density-to-rms-noise}
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From ([Fleming 2010](#orgf17a758)):
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\\[ \text{RMS noise} = \sqrt{2 \times \text{bandwidth}} \times \text{noise density} \\]
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If the noise is normally distributed, the RMS value is also the standard deviation \\(\sigma\\).
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The peak to peak amplitude is then approximately \\(6 \sigma\\).
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<div class="exampl">
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<div></div>
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- noise density = \\(20 pm/\sqrt{Hz}\\)
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- bandwidth = 100Hz
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\\[ \sigma = \sqrt{2 \times 100} \times 20 = 0.28nm RMS \\]
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The peak-to-peak noise will be approximately \\(6 \sigma = 1.7 nm\\)
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</div>
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## Bibliography {#bibliography}
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<a id="orgf17a758"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
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<a id="orgf2f4e47"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.
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